Improved minimax predictive densities under Kullback-Leibler loss
Improved minimax predictive densities under Kullback-Leibler loss
复制标题
改进 Kullback-Leibler 损失下的极小极大预测密度
DOI：
10.1214/009053606000000155
复制
发表时间：
2006
影响因子：
4.5
通讯作者：
Xinyi Xu
中科院分区：
文献类型：
--
作者：
E. George;Feng Liang;Xinyi Xu
令x | μ〜n p（μ，u x i）和y | μ〜 n p（μ，v y i）为独立的多维多变量正常向量，基于观察到x = x，我们考虑了常见的平均值。这个问题的自然程序是（正式的）贝叶斯预测密度p u（y | x）在均匀的先验πu（μ）下1是最不变的，而我们表明，如果先前的边缘是超谐的，或者其平方根是超级谐波，则任何贝叶斯预测密度将是最小值。 U（Y | X），包括贝叶斯的预测密度描述了二次损失。
Let X|μ ∼ N p (μ, u x I) and Y|μ ∼ N p (μ, v y I) be independentp-dimensional multivariate normal vectors with common unknown mean μ. Based on only observing X = x, we consider the problem of obtaining a predictive density p(y|x) for Y that is close to p(y|μ) as measured by expected Kullback-Leibler loss. A natural procedure for this problem is the (formal) Bayes predictive density p U (y|x) under the uniform prior π U (μ) ≡ 1, which is best invariant and minimax. We show that any Bayes predictive density will be minimax if it is obtained by a prior yielding a marginal that is superharmonic or whose square root is superharmonic. This yields wide classes of minimax procedures that dominate p U (y|x), including Bayes predictive densities under superharmonic priors. Fundamental similarities and differences with the parallel theory of estimating a multivariate normal mean under quadratic loss are described.